Let `0 lt alpha lt (pi)/(2) ` be a fixed angle . If `p=(costheta, sin theta) and Q(cos(alpha-theta))`, then Q is obtained from P by

Options

A. (a)clockwise rotation around origin through an angle `alpha`
B. (b)anit-clockwise rotation around origin through an angle `alpha`
C. (c)reflection in the line through origin with slope `tanalpha`
D. (d)reflection in the line through origin which slope `tan((alpha)/(2))`

Answer

To solve the problem, we need to analyze the relationship between the points P and Q given the angle α and the coordinates of P and Q.

### Step-by-Step Solution:

1. **Identify the Coordinates of P and Q**:
- The point P is given as \( P = (\cos \theta, \sin \theta) \).
- The point Q is given as \( Q = (\cos(\alpha - \theta), \sin(\alpha - \theta)) \).

2. **Understand the Geometric Representation**:
- The point P lies on the unit circle at an angle \( \theta \) from the positive x-axis.
- The point Q lies on the unit circle at an angle \( \alpha - \theta \) from the positive x-axis.

3. **Determine the Angle Between P and Q**:
- The angle between the position vector OP (from the origin O to point P) and OQ (from the origin O to point Q) is \( \alpha \).
- This is because \( OQ \) is at an angle \( \alpha - \theta \) and \( OP \) is at an angle \( \theta \).

4. **Rotation from P to Q**:
- To go from point P to point Q, we need to rotate the vector OP by the angle \( \alpha \) in the clockwise direction.
- Hence, Q is obtained from P by a clockwise rotation around the origin through the angle \( \alpha \).

5. **Conclusion**:
- Therefore, the final answer is that Q is obtained from P by a clockwise rotation through the angle \( \alpha \) around the origin.